NIntegrate and speed

*To*: mathgroup at smc.vnet.net*Subject*: [mg116780] NIntegrate and speed*From*: Marco Masi <marco.masi at ymail.com>*Date*: Sun, 27 Feb 2011 04:35:46 -0500 (EST)

I have the following problems with NIntegrate. 1) I would like to make the following double numerical integral converge without errors R = 8000; Z = 1; rd = 3500; NIntegrate[Exp[-k Abs[Z]]/(1 + (k rd)^2)^1.5 (NIntegrate[Cos[k R Sin[\[Theta]]], {\[Theta], 0, \[Pi]}]), {k, 0, \[Infinity]}] It tells non numerical values present and I don't understand why, since it evaluates finally a numerical value? 0.000424067 2) Isn't the second integrand a cylindrical Bessel function of order 0? So, I expected that NIntegrate[Exp[-k Abs[Z]]/(1 + (k rd)^2)^1.5 BesselJZero[0, k R], {k, 0, \[Infinity]}] doing the same job. But it fails to converge and gives 0.00185584- i4.96939*10^-18. Trying with WorkingPrecision didn't make things better. How can this be fixed? 3) The above Nintegrals will go into a loop and should be evaluated as fast as possible. How? With Compile, CompilationTarget -> "C", Paralleization, etc.? Any suggestions? Marco.